Bifurcations of travelling wave solutions in a new integrable equation with peakon and compactons |
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| Affiliation: | 1. College of Science, P. O. Box 253, University of Shanghai for Science and Technology, Shanghai 200093, China;2. Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA;3. Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China;4. College of Mathematics and Systems Science, Shandong University of Science and Technology,Qingdao, Shandong 266590, China;5. International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Mmabatho 2735, South Africa |
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| Abstract: | Degasperis and Procesi applied the method of asymptotic integrability and obtain Degasperis–Procesi equation. They showed that it has peakon solutions, which has a discontinuous first derivative at the wave peak, but they did not explain the reason that the peakon solution arises. In this paper, we study these non-smooth solutions of the generalized Degasperis–Procesi equation ut − utxx + (b + 1)uux = buxuxx + uuxxx, show the reason that the non-smooth travelling wave arise and investigate global dynamical behavior and obtain the parameter condition under which peakon, compacton and another travelling wave solutions engender. Under some parameter condition, this equation has infinitely many compacton solutions. Finally, we give some explicit expression of peakon and compacton solutions. |
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